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Entropy 2018, 20(4), 220; https://doi.org/10.3390/e20040220

Logarithmic Sobolev Inequality and Exponential Convergence of a Markovian Semigroup in the Zygmund Space

Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Received: 29 December 2017 / Revised: 19 March 2018 / Accepted: 19 March 2018 / Published: 23 March 2018
(This article belongs to the Special Issue Entropy and Information Inequalities)
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Abstract

We investigate the exponential convergence of a Markovian semigroup in the Zygmund space under the assumption of logarithmic Sobolev inequality. We show that the convergence rate is greater than the logarithmic Sobolev constant. To do this, we use the notion of entropy. We also give an example of a Laguerre operator. We determine the spectrum in the Orlicz space and discuss the relation between the logarithmic Sobolev constant and the spectral gap. View Full-Text
Keywords: Dirichlet form; logarithmic Sobolev inequality; entropy; spectrum; Zygmund space; Laguerre operator Dirichlet form; logarithmic Sobolev inequality; entropy; spectrum; Zygmund space; Laguerre operator
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Shigekawa, I. Logarithmic Sobolev Inequality and Exponential Convergence of a Markovian Semigroup in the Zygmund Space. Entropy 2018, 20, 220.

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